Penner's theorem

Penner's theorem establishes a direct link between geometric transformations known as Dehn twists on a surface and their algebraic representations in the surface's mapping class group. Specifically, it states that any composition of Dehn twists can be expressed through a sequence of elementary operations, and the entire group of these transformations has a structure that can be understood in algebraic terms. This helps mathematicians analyze complex surface symmetries by breaking them down into simple, well-understood twists, providing a bridge between geometric intuition and algebraic formalism.